Abstract
In this paper, we consider a discrete formulation of the one-dimensional \(L^1pTV\) functional and introduce a finite algorithm that finds exact minimizers of this functional for \(0<p\le 1\). Our algorithm for the special case for \(L^1TV\) returns globally optimal solutions for all regularization parameters \(\lambda \ge 0\) at the same computational cost of determining a single optimal solution associated with a particular value of \(\lambda \). This finite set of minimizers contains the scale signature of the known initial data. A variation on this algorithm returns locally optimal solutions for all \(\lambda \ge 0\) for the case when \(0<p<1\). The algorithm utilizes the geometric structure of the set of hyperplanes defined by the nonsmooth points of the \(L^1pTV\) functional. We discuss efficient implementations of the algorithm for both general and binary data.
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Notes
The sunspot number is commonly refered to as the Wolf Number in honor of Rudolf Wolf who is credited with the concept in 1848.
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Acknowledgments
We would like to acknowledge Jamie O’Brien for finding a mistake in our algorithm description. We would also like to thank the reviewers for taking the time to give valuable feedback on our original manuscript.
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Moon, H.A., Asaki, T.J. A finite hyperplane traversal Algorithm for 1-dimensional \(L^1pTV\) minimization, for \(0<p\le 1\) . Comput Optim Appl 61, 783–818 (2015). https://doi.org/10.1007/s10589-015-9738-4
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DOI: https://doi.org/10.1007/s10589-015-9738-4